I can't calculate it. Everything I try leads to difficult integrals. Could you please give me an idea of what the right way of doing this is?

The condition on z for a point to lie in U is , where The integral for negative z is the same as for positive z, and so

(doing the z integral first).

The variables conveniently separate and give you Check that For the other integral, substitute and get the integral in the form

which you can easily integrate. I get the answer for the measure of U to be (but don't trust my accuracy there). Note that this becomes infinite when k=2. I guess that U must have infinite measure whenever .